{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Black Scholes valuation methods\n",
    "\n",
    "[*Black Scholes model WIKI*](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model). \n",
    "\n",
    "\n",
    "The purpose of this notebook is to review the most common algorithms and implement them numerically. \n",
    "\n",
    "## Contents\n",
    "   - [European option](#sec1)\n",
    "      - [Put-Call parity](#sec1.1)\n",
    "   - [Numerical integration](#sec2)\n",
    "   - [Monte Carlo method](#sec3)\n",
    "   - [Binomial tree](#sec4)\n",
    "   - [Limits of the model](#sec5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from functions.BS_pricer import BS_pricer\n",
    "from functions.Parameters import Option_param\n",
    "from functions.Processes import Diffusion_process\n",
    "\n",
    "import numpy as np\n",
    "import scipy as scp\n",
    "import scipy.stats as ss\n",
    "from scipy.integrate import quad\n",
    "from functools import partial\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec1'></a>\n",
    "## European option\n",
    "\n",
    "Under the Black-Scholes (BS) model, the best method to price a vanilla European option is to use the [BS closed formula](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model).\n",
    "\n",
    "The **BS formula** for a call is:\n",
    "\n",
    "$$ C(t,T,S_t,K,r,\\sigma) = S_t N(d_1) - K e^{-r(T-t)} N(d_2) $$\n",
    "\n",
    "with \n",
    "\n",
    "$$ d_1 = \\frac{1}{\\sigma \\sqrt{T-t}} \\biggl[ \\log \\biggl( \\frac{S_t}{K} \\biggr) + \\biggl(r + \\frac{\\sigma^2}{2} \\biggr) (T-t) \\biggr] \\quad \\mbox{and} \\quad d_2 = d_1 - \\sigma \\sqrt{T-t} $$\n",
    "\n",
    "where $N$ is the cumulative distribution function of a standard normal random variable.    \n",
    "The formula for a put is similar and can be found in the wiki page.\n",
    "\n",
    "The value of an option can be also computed as the discounted expectation of a future payoff in this way:\n",
    "\n",
    "$$\\begin{aligned}\n",
    " C(S_t,K,T) &= e^{-r(T-t)} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ (S_T - K)^+ \\bigg| S_t \\biggr] \\\\\n",
    "            &= e^{-r(T-t)} \\int_0^{\\infty} (S_T - K)^+ f(S_T|S_t) dS_T   \\quad \\text{(see comment below)} \\\\\n",
    "            &= e^{-r(T-t)} \\int_K^{\\infty} (S_T - K) f(S_T|S_t) dS_T \n",
    "\\end{aligned}$$\n",
    "\n",
    "where $f(S_T|S_t)$ is the risk neutral transition probability of the process $\\{S_u\\}_{u\\in [t,T]}$. This is a log-normal density function \n",
    "\n",
    "$$ f(S_T|S_t) = \\frac{1}{S_T \\sigma \\sqrt{2\\pi (T-t)}} \\; e^{- \\frac{ \\biggl[\\log(S_T) - \\bigl(\\log(S_t) + (r-\\frac{1}{2} \\sigma^2)(T-t) \\bigr) \\biggr]^2}{2\\sigma^2 (T-t)}} $$\n",
    "\n",
    "\n",
    "#### Comment:\n",
    "In this context there is a clear abuse of notation!! Usually in statistics random variables are indicated by capital letters, and non-random variables are indicated with small letters. However, it is very common to indicate the stock price in the Black-Scholes formula with a capital letter. Here I used this notation in order to be consistent with the [wiki notation](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model#Notation), although I think this choice can generate misunderstandings. To summarize:    \n",
    "In the function $C(S_t,K,T)$ the variables are non-random! Inside the expectation $\\mathbb{E}^{\\mathbb{Q}}\\bigl[ (S_T - K)^+ \\big| S_t \\bigr]$, the variable $S_T$ is random. In the second and third lines, $S_T$ is not random and consequently the integral in $dS_T$ **is NOT** a stochastic integral. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let us analyze better the previous formulas\n",
    "\n",
    "\n",
    "\n",
    "\\begin{align*}\n",
    " C(S_t,K,T) &= e^{-r(T-t)} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ (S_T - K)^+ \\bigg| S_t \\biggr] \\\\\n",
    "            &= e^{-r(T-t)} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ S_T \\mathbb{1}_{S_T >K} \\bigg| S_t \\biggr] \n",
    "              - e^{-r(T-t)} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ K \\mathbb{1}_{S_T >K} \\bigg| S_t \\biggr] \\\\\n",
    "            &= e^{-r(T-t)} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ S_T \\mathbb{1}_{S_T >K} \\bigg| S_t \\biggr] \n",
    "              - e^{-r(T-t)} K \\, \\underbrace{\\mathbb{Q}\\biggl[ S_T >K \\bigg| S_t \\biggr]}_{N(d_2)} \\\\            \n",
    "\\end{align*}\n",
    "\n",
    "Let us introduce the following change of measure (under the stock numeraire):\n",
    "\n",
    "$$ \\begin{aligned}\n",
    "\\frac{d \\tilde{\\mathbb{Q}} }{ d \\mathbb{Q} } &= \\frac{S_T}{\\mathbb{E}^\\mathbb{Q}[S_T]} = \\frac{S_T}{S_t e^{r(T-t)}}  \\\\ \n",
    "                                            &= \\frac{S_t e^{(r -\\frac{1}{2}\\sigma^2)(T-t) + \\sigma W_{T-t}} }{S_t e^{r(T-t)}} \\\\\n",
    "                                            &=   e^{ -\\frac{1}{2}\\sigma^2(T-t) + \\sigma W_{T-t} } \\quad \\text{(exponential martingale)} \n",
    "\\end{aligned} $$\n",
    "\n",
    "By [Girsanov theorem](https://en.wikipedia.org/wiki/Girsanov_theorem), under $\\tilde{\\mathbb{Q}}$ the driving Brownian motion has the new dynamics \n",
    "\n",
    "$$ \\tilde{W_t} = W_t - \\sigma t $$\n",
    "\n",
    "and the corresponding stock dynamics becomes\n",
    "\n",
    "$$\\begin{aligned}\n",
    " \\frac{dS_t}{S_t} &= r dt + \\sigma dW_t \\\\\n",
    "                  &= (r+\\sigma^2) dt + \\sigma d\\tilde{W}_t \n",
    "\\end{aligned}$$\n",
    "\n",
    "The first term is\n",
    "\n",
    "$$ \\begin{aligned}\n",
    " e^{-r(T-t)} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ S_T \\mathbb{1}_{S_T >K} \\bigg| S_t \\biggr] =& e^{-r(T-t)} \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \n",
    "                 \\biggl[ \\frac{d \\mathbb{Q} }{ d \\tilde{\\mathbb{Q}}}  S_T \\mathbb{1}_{S_T >K} \\bigg| S_t \\biggr] \\\\\n",
    "                     =& e^{-r(T-t)} \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \n",
    "                 \\biggl[ \\frac{e^{r(T-t)}S_t}{S_T}  S_T \\mathbb{1}_{S_T >K} \\bigg| S_t \\biggr] \\\\ \n",
    "                    =& S_t \\, \\underbrace{\\tilde{\\mathbb{Q}} ( S_T > K | S_t)}_{N(d_1)}\n",
    "\\end{aligned}$$\n",
    "\n",
    "We have just seen how to interpret the terms $N(d_1)$ and $N(d_2)$. These are the risk neutral probabilities of $S_T > K$ in the stock and money market numeraires respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I implemented the BS closed formula in the class `BS_pricer`.     \n",
    "Let us consider the following set of parameters, that will be recurrent in all the next notebooks. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "S0=100.0    # spot stock price\n",
    "K=100.0     # strike\n",
    "T=1.0       # maturity \n",
    "r=0.1       # risk free rate \n",
    "sig=0.2     # diffusion coefficient or volatility"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Call price:  13.269676584660893\n",
      "Put price:  3.753418388256833\n"
     ]
    }
   ],
   "source": [
    "call = BS_pricer.BlackScholes(\"call\",S0,K,T,r,sig) \n",
    "put = BS_pricer.BlackScholes(\"put\",S0,K,T,r,sig) \n",
    "print(\"Call price: \", call )  \n",
    "print(\"Put price: \", put ) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec1.1'></a>\n",
    "### Put-Call parity\n",
    "\n",
    "This is an important formula [wiki page](https://en.wikipedia.org/wiki/Put%E2%80%93call_parity)\n",
    "\n",
    "$$ Call - Put = S_0 - K e^{-rT}  $$\n",
    "\n",
    "Let us check if it works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "13.269676584660893\n",
      "13.269676584660886\n"
     ]
    }
   ],
   "source": [
    "print(call) \n",
    "print(put + S0 - K * np.exp(-r*T) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2'></a>\n",
    "## Numerical integration\n",
    "\n",
    "I want to play a bit with the different formulas written above.  \n",
    "\n",
    "Let us compute the option prices by integrating the log-normal density:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "e_ret = np.log(S0) + ( r - 0.5 * sig**2 ) * T   # expected return of the log-price\n",
    "vol = sig * np.sqrt(T)                          # standard deviation of the log-price\n",
    "\n",
    "# log-normal density (defined above) \n",
    "def log_normal(x, e_ret, vol):\n",
    "    return 1/(x*vol*np.sqrt(2*np.pi)) * np.exp(- (np.log(x) - e_ret)**2 /(2*vol**2) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 576x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8,5))\n",
    "x = np.linspace(30,190, 100)\n",
    "plt.plot(x, log_normal(x, e_ret,vol))\n",
    "plt.title(\"Lognormal distribution, conditioned on $S_0=100$\")\n",
    "plt.xlabel(\"$S_T$\"); plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function `log_normal(x, e_ret, vol)` defined above, corresponds to the scipy.stats function: \n",
    "```\n",
    "ss.lognorm.pdf(x, vol, scale=np.exp(e_ret) ).\n",
    "```\n",
    "\n",
    "In the next calculation, I'm going to use the scipy function.    \n",
    "Let us perform the integration with the `scipy.integrate` function [quad](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Call price: 13.269676584660926 \n",
      "Put price: 3.753418388256828 \n"
     ]
    }
   ],
   "source": [
    "def integrand_LN(S, strike, e_ret, vol, payoff):\n",
    "    if payoff == \"call\":\n",
    "        return (S - strike ) * ss.lognorm.pdf(S, vol, scale=np.exp(e_ret))\n",
    "    elif payoff == \"put\":\n",
    "        return (strike - S) * ss.lognorm.pdf(S, vol, scale=np.exp(e_ret))\n",
    "\n",
    "call = quad(integrand_LN, K, np.inf, args=(K, e_ret, vol, \"call\") )[0]  * np.exp(-r*T)\n",
    "put = quad(integrand_LN, 0, K, args=(K, e_ret, vol, \"put\") )[0]  * np.exp(-r*T)\n",
    "\n",
    "print(\"Call price: {} \\nPut price: {} \".format(call,put) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The put option payoff $(K-S_T)^+$ is positive for $S_T < K$.    \n",
    "- In the call case, the integration is from $K$ to $\\infty$.\n",
    "- In the put case, the integration is from $0$ to $K$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if we use the change of measure proposed above?  In this way the integrations are simpler.    \n",
    "Let us compute $\\tilde{\\mathbb{Q}}( S_T > K )$ and $\\mathbb{Q}( S_T > K )$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Risk neutral probability under stock numeraire,\n",
      " Q1 = 0.7257468822499274\n",
      "Risk neutral probability under money market numeraire,\n",
      " Q2 = 0.6554217416103069\n",
      "BS call price:  13.269676584662555\n"
     ]
    }
   ],
   "source": [
    "e_ret_1 = np.log(S0) + ( r + 0.5 * sig**2 ) * T  # expected return of the log-price under the new measure\n",
    "\n",
    "Q1 = quad(lambda S: ss.lognorm.pdf(S, vol, scale=np.exp(e_ret_1)), K, np.inf )[0]\n",
    "print(\"Risk neutral probability under stock numeraire,\\n Q1 =\", Q1)\n",
    "Q2 = quad(lambda S: ss.lognorm.pdf(S, vol, scale=np.exp(e_ret)), K, np.inf )[0]\n",
    "print(\"Risk neutral probability under money market numeraire,\\n Q2 =\", Q2)\n",
    "\n",
    "print(\"BS call price: \", S0 * Q1 - K* np.exp(-r*T) *Q2  )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is quite common to compute the Black-Scholes formula using $N(d_1)$ and $N(d_2)$.   \n",
    "The reason is that the cumulative function of the standard Normal distribution is more accessible (I guess). In the `BS_pricer` class I used the function `scipy.stats.norm.cdf`.     \n",
    "\n",
    "For completeness, let me recall that if $X_T$ is a Normal random variable, then $S_T = S_0 e^{X_T}$ is Log-Normal. Therefore we have:\n",
    "\n",
    "$$ \\mathbb{Q}( S_T > K ) = \\mathbb{Q}\\biggl( S_0 e^{X_T} > K \\biggr) = \\mathbb{Q}\\biggl( X_T > \\log \\frac{K}{S_0} \\biggr). $$\n",
    "\n",
    "This permits to use the Normal cumulative function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3'></a>\n",
    "## Monte Carlo method\n",
    "\n",
    "I'm going to simulate the random variables: \n",
    "\n",
    "$$ S_T^i = S_0 e^{(r -\\frac{1}{2}\\sigma^2)T + \\sigma W_{T}^i} $$\n",
    "\n",
    "for $1 \\leq i \\leq N$.    \n",
    "Then use the approximation for a call option:\n",
    "\n",
    "$$ \\mathbb{E}^{\\mathbb{Q}}\\biggl[ (S_T - K)^+ \\bigg| S_0 \\biggr] \\; \n",
    "\\approx \\; \\frac{1}{N} \\sum_{i=1}^N (S_T^i - K)^+\n",
    "$$\n",
    "\n",
    "For a put option I use this payoff $(K - S_T )^+$ inside the expectation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(seed=44)  # seed for random number generation\n",
    "N = 10000000  # Number of random variables\n",
    "\n",
    "W = ss.norm.rvs( (r-0.5*sig**2)*T , np.sqrt(T)*sig, N)\n",
    "S_T = S0 * np.exp(W)\n",
    "\n",
    "call = scp.mean( np.exp(-r*T) * np.maximum(S_T-K,0) )\n",
    "put = scp.mean( np.exp(-r*T) * np.maximum(K-S_T,0) )\n",
    "call_err = ss.sem( np.exp(-r*T) * np.maximum(S_T-K,0) )  # standard error\n",
    "put_err = ss.sem( np.exp(-r*T) * np.maximum(K-S_T,0) )   # standard error"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Call price: 13.26333800663666, with error: 0.005093638687208466\n",
      "Put price: 3.7553894006350093, with error: 0.002214066662789331\n"
     ]
    }
   ],
   "source": [
    "print(\"Call price: {}, with error: {}\".format(call, call_err))\n",
    "print(\"Put price: {}, with error: {}\".format(put, put_err))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### BS_pricer\n",
    "\n",
    "In the next notebooks I will present better the class `BS_pricer`. But now let's have a look at the prices obtained by different pricing methods:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creates the object with the parameters of the option\n",
    "opt_param = Option_param(S0=100, K=100, T=1, exercise=\"European\", payoff=\"call\" )\n",
    "# Creates the object with the parameters of the process\n",
    "diff_param = Diffusion_process(r=0.1, sig=0.2)\n",
    "# Creates the pricer object\n",
    "BS = BS_pricer(opt_param, diff_param)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.269676584660893"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BS.closed_formula()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.2696765846609"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BS.Fourier_inversion()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(13.269912012325573, 0.0029413719629421543, 1.9152190685272217)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BS.MC(N=30000000, Err=True, Time=True)  # output is: price, standard error and execution time "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "BS.mesh_plt()   # PDE method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The PDE approach is the topic of the notebook **2.1**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec4'></a>\n",
    "## Binomial tree\n",
    "\n",
    "\n",
    "Of course I cannot forget about the Binomial model!\n",
    "This is a simple but very powerful numerical method!\n",
    "\n",
    "I expect you to be familiar with this model. If not, have a look at the [wiki page](https://en.wikipedia.org/wiki/Binomial_options_pricing_model). \n",
    "Although I said I expect you to know the model, I'm not expecting that you have already implemented it!     \n",
    "Therefore, here I present an efficient implementation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "BS Tree Price:  13.269537371978052\n"
     ]
    }
   ],
   "source": [
    "N = 15000              # number of periods or number of time steps  \n",
    "payoff = \"call\"        # payoff \n",
    "\n",
    "dT = float(T) / N                             # Delta t\n",
    "u = np.exp(sig * np.sqrt(dT))                 # up factor\n",
    "d = 1.0 / u                                   # down factor \n",
    "\n",
    "V = np.zeros(N+1)                             # initialize the price vector\n",
    "S_T = np.array( [(S0 * u**j * d**(N - j)) for j in range(N + 1)] )  # price S_T at time T\n",
    "\n",
    "a = np.exp(r * dT)    # risk free compounded return\n",
    "p = (a - d)/ (u - d)  # risk neutral up probability\n",
    "q = 1.0 - p           # risk neutral down probability   \n",
    "\n",
    "if payoff ==\"call\":\n",
    "    V[:] = np.maximum(S_T-K, 0.0)\n",
    "else:\n",
    "    V[:] = np.maximum(K-S_T, 0.0)\n",
    "\n",
    "for i in range(N-1, -1, -1):\n",
    "    V[:-1] = np.exp(-r*dT) * (p * V[1:] + q * V[:-1])    # the price vector is overwritten at each step\n",
    "        \n",
    "print(\"BS Tree Price: \", V[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec5'></a>\n",
    "## Limits of the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "BS_sigma = partial(BS_pricer.BlackScholes, \"call\", S0, K, T, r) # binding the function \n",
    "sigmas = np.linspace(0.01, 10, 1000)\n",
    "\n",
    "plt.plot(sigmas, BS_sigma(sigmas))\n",
    "plt.xlabel(\"sig\"); plt.ylabel(\"price\"); plt.title(\"Black-Scholes price as function of volatility\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The BS formula is an increasing function of the volatility.    \n",
    "However, for higher volatilities, the graph becomes almost flat!!\n",
    "\n",
    "We can conclude that the model is reliable for volatilities in the range $0 - 400\\%$.\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
